| Pfaffian state | |
|---|---|
| Name | Pfaffian state |
| Associated with | Fractional quantum Hall effect |
| Introduced by | Gregory Moore and Nicholas Read |
| Year | 1991 |
| Notable for | Non‑Abelian anyons, trial wavefunction for filling factor 5/2 |
| Applications | Topological quantum computation |
Pfaffian state
The Pfaffian state is a proposed quantum many‑body state of two‑dimensional electrons that captures pairing correlations in certain fractional quantum Hall regimes. It is notable for supporting non‑Abelian quasiparticle excitations and providing a concrete platform for proposals in topological quantum computation. The state connects condensed matter physics, algebraic structures in mathematics, and experimental research in semiconductor heterostructures and graphene.
The Pfaffian state was introduced in the context of the even‑denominator plateau observed at filling factor ν = 5/2 in high‑mobility GaAs/AlGaAs heterostructures. Its importance lies in explaining a puzzling incompressible quantum Hall state that cannot be accounted for by Abelian composite fermion theory alone. The state emerges from strong magnetic fields and low temperatures where electrons are confined to the lowest few Landau levels, and electron–electron interactions dominate kinetic energy. The concept ties to the broader study of quantum many‑body systems, topological phases of matter, and low‑dimensional electron gases studied at institutions such as the Bell Labs, IBM, and university laboratories.
Mathematically, the Pfaffian state is represented by a trial wavefunction constructed from a Pfaffian of pairwise correlators multiplied by a Jastrow factor. The canonical form for N electrons in the lowest Landau level is Ψ_Pf = Pf(1/(z_i - z_j)) ∏_{i
In the fractional quantum Hall effect (FQHE), the Pfaffian provides a theoretical candidate for the incompressible phase at ν = 5/2 and potentially other even‑denominator fractions. It differs from Laughlin and composite fermion states by encoding p‑wave pairing of composite fermions, producing a gapped bulk with chiral edge modes described by an Ising conformal field theory plus a charged boson. The Pfaffian and its particle‑hole conjugate, the anti‑Pfaffian, represent competing topological orders consistent with experiments measuring thermal conductance and edge transport. Understanding which topological order is realized bears on microscopic Hamiltonians such as the Coulomb interaction projected into the second Landau level and model Hamiltonians with three‑body interactions introduced to stabilize Pfaffian correlations.
A defining feature of the Pfaffian state is its quasiparticle excitations obeying non‑Abelian braid statistics; exchanging two quasiparticles transforms the ground‑state manifold by a unitary matrix rather than a scalar phase. These non‑Abelian anyons are associated with Majorana zero modes and Ising‑type fusion rules that were first noted in the Moore–Read construction. The non‑Abelian nature makes the Pfaffian attractive for Topological quantum computation proposals, where braiding and fusion provide fault‑tolerant logical gates. Institutions and projects such as Microsoft Quantum have cited non‑Abelian quantum Hall states among candidate platforms for topological qubits, though Ising anyons alone do not provide a universal gate set without supplementing operations like magic state distillation.
Experimental support for Pfaffian physics arises from transport, tunneling, and thermal conductance measurements in high‑mobility GaAs/AlGaAs heterostructures and other two‑dimensional electron systems, including recent studies in ZnO and graphene where even‑denominator states have been reported. Key signatures include quantized Hall plateaus at ν = 5/2, quasiparticle charge e/4 detected in shot noise and tunneling experiments, upstream neutral modes on edges, and quantized thermal Hall conductance consistent with predicted chiral central charge. Interferometry experiments, such as Fabry–Pérot and Mach–Zehnder setups pursued at university and national laboratories, aim to detect non‑Abelian braiding directly, but interpretations have been complicated by disorder, edge reconstruction, and equilibration effects.
Theoretical efforts continue to refine the microscopic stability of Pfaffian order, examining particle‑hole symmetry, Landau level mixing, disorder, and realistic interaction potentials. The anti‑Pfaffian, a particle‑hole conjugate state with opposite edge chirality, competes energetically under certain conditions; hybridized and striped phases, composite fermion liquid states, and other paired states (e.g., the 331 state) are considered viable alternatives. Numerical studies using exact diagonalization, density‑matrix renormalization group (DMRG), and matrix product states at institutions like Princeton University, Stanford University, and national supercomputing centers test candidate Hamiltonians. The Pfaffian remains a central concept for understanding topological order in two dimensions and for guiding experimental searches that emphasize stability, coherence, and national research coordination in quantum materials and quantum information science.
Category:Quantum Hall states Category:Topological phases of matter Category:Non‑Abelian anyons